Georg P. Mueller – Faculty of Economics and Social Sciences, University of Fribourg, Fribourg, Switzerland

Part of the Book: Advances in Mathematics Research. Volume 36

Abstract

This article deals with the use of modal and fuzzy logic for analyzing the content validity of indicators. It is based on the assumption that the estimated truth of the presence of a concept in a given social situation depends on the indicator that is used for its representation. The respective truth is fuzzy and lies on a continuum from true to false. Between these two poles there is a sigmoid fuzzy truth function, which can be determined by a logistic regression of big survey data that explains the presence or absence of the concept by the mentioned indicator. The resulting logistic function generally has three areas: one near the pole 1 = true, where the indicator values stand for the presence of the concept. In terms of modal logic, the presence of the concept is necessary. A second area is near the pole 0 = false, where the indicator values point with high certainty to the absence of the concept. Thus, in terms of modal logic its non-presence is necessary. Finally, there is a third area in between, where the analyzed concept is only possible. Any knowledge gains based on indicator values close to the poles with fuzzy truth 0 or 1 have a high validity with regard to the related concept. Thus, the first two areas should be as large as possible and in this way limit the area of indicator values with indeterminate truth. Hence, when deciding between alternative indicators, the geometries of the logistic functions are as important as the r-squares of the related regressions. Equivalent indicators can also be combined by Boolean operators like AND, OR, etc. For illustrative purposes, the author explores the popular views of the concept of religiosity in the European Values Study EVS-2008: the frequencies of church-going and private prayers are compared with regard to their validity as operationalizations of the mentioned concept.

Keywords: modal logic, fuzzy truth, validity of indicators, big surveys, alternative operationalizations, logistic regression

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